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This paper deals with the geometry of the space [Geometric Invariant Theory (GIT) quotient] M8 of eight points in ℙ1, and the Gale-quotient N′8 of the GIT quotient of eight points in ℙ3.The space M8 comes with a natural embedding in ℙ13, or more precisely, the projectivization of the 8-representation V4, 4. There is a single 8-skew cubic  in ℙ13. The fact that M8 lies on the skew cubic  is a consequence of Thomae's formula for hyperelliptic curves, but more is true: M8 is the singular locus of . These constructions yield the free resolution of M8 and are used in the determination of the ‘single’ equation cutting out the GIT quotient of n points in ℙ1 in general [20].The space N′8 comes with a natural embedding in ℙ13, or more precisely, ℙ V2, 2, 2, 2. There is a single skew quintic  containing N′8 and N′8 is the singular locus of the skew quintic .The skew cubic  and skew quintic  are projectively dual. (In particular, they are surprisingly singular, in the sense of having a dual of remarkably low degree.) The divisor on the skew cubic blown down by the dual map is the secant variety Sec(M8) and the contraction Sec(M8) → N′8 factors through N8 via the space of eight points on a quadric surface. We conjecture (Conjecture 1.1) that the divisor on the skew quintic blown down by the dual map is the quadrisecant variety of N′8 (the closure of the union of quadrisecant lines), and that the quintic  is the trisecant variety. The resulting picture extends the classical duality in the 6-point case between the Segre cubic threefold and the Igusa quartic threefold.We note that there are a number of geometrically natural varieties that are (related to) the singular loci of remarkably singular cubic hypersurfaces (see for example [3, 5]). [ABSTRACT FROM PUBLISHER]